Despite these important pioneering works, we consider the more modern notion of NC geometry to begin in earnest with the work ² of [18–20], which provides an axiomatic definition of Newtonian gravity.

In all three articles, Trautman gave an axiomatic definition of NC gravity. The precise set of postulates changes slightly from one article to another. We follow [20], where Trautman introduced the following postulates:

(1) Spacetime is a four-dimensional differentiable manifold endowed with a symmetric affine connection Γ μ ν ρ .

As the connection is symmetric, taking the antisymmetric part of ∇ _μ τ _ν = 0 implies that 2 ∂ μ τ ν = 0 , or equivalently dτ = 0 in form notation. The first of the two conditions in (2.2) was later dropped in favor of a boundary condition, as discussed in the following equations. The second condition in (2.2) is often simply referred to as the Trautman condition. In various places in the literature, it is phrased as h λ [ μ R λ ρ σ ν ] = 0 . (2.4)

The latter is equivalent to R μ ρ ν σ = R μ ρ σ ν , (2.5) which appears to be weaker than the second condition of (2.2). However, using ∇ _ρ h ^μν = 0, which implies [∇ _ρ , ∇ _σ ]h ^μν = 0 or equivalently h κ ν R ρ σ κ μ + h κ μ R ρ σ κ ν = 0 , Eq. 2.4 can be equivalent to the second condition of (2.2).

Around the time of Trautman’s work, the mathematical framework underlying NC geometry was put on a firm foundation in [21] (see also the later work by [22]). We will not focus on the mathematical development of NC geometry in this review, so we will not comment further on these works.

Trautman’s work can be viewed as an intrinsic definition of Newtonian gravity without any reference to 1/c expansions of GR. It was later improved by [22], who, among other things, worked out the class of torsion-free metric-compatible connections and realized that the first Trautman condition (the first equation in (2.2) is redundant if we assume asymptotic flatness. It was shown by [23, 24] that an appropriate covariant expansion of GR in powers of 1/c reproduces Trautman’s formulation of NC gravity, with the caveat that the first Trautman condition is dropped. We refer to Sections 6 Sections 3.8 for more details. The first Trautman condition is equivalent to the following condition: h ρ κ R μ ν ρ σ = 0 that Dautcourt in [24] attributed to Dixon ³ [25]. Dautcourt wrote that this condition did not follow from the 1/c expansion of GR. Conversely, the second equation of (2.2) does follow from the 1/c expansion of the Riemann tensor of GR. More details will be given in the following paragraphs.

In [24], Dautcourt only discussed the expansion in terms of even powers of 1/c. From the outset, the NC connection is considered symmetric and metric-compatible. As shown previously, this implies that dτ = 0 so that ⁴ τ = dT for a time coordinate T. Therefore, time is absolute in this notion of NC geometry. This is an important restriction from the 1/c expansion perspective. Dautcourt knew that the latter could be formulated without this restriction but did not consider the general case.

Later, Dautcourt [26] included odd powers in 1/c. Dautcourt more explicitly discussed the option of allowing for a non-trivial lapse function N in NC geometry so that τ = NdT, which is now known as twistless torsional NC (TTNC) geometry. It is equivalent to the Frobenius condition τ ∧ dτ = 0, implying that equal T surfaces define spatial hypersurfaces. ⁵ In particular, Dautcourt’s concluded that the 1/c expansion of GR leads to a theory that is more general than Newtonian gravity (on page 7) but then suggested that this more general case with N a priori arbitrary is not so interesting because insisting on global regularity of the NC lapse function N would reduce N to a constant and thus force time to be absolute. The argument here is that the NC lapse function is harmonic, and using Liouville’s theorem, it must be constant to be regular everywhere without appropriate sources. The current viewpoint is that the condition of global regularity of the NC lapse function is too restrictive because this rules out interesting non-relativistic approximations of GR (given in the following sections) and that one can find sources leading to singularities in N.

Ehlers [27, 28] developed “frame theory,” which is a geometrical formulation that treats Lorentzian and Galilean geometry on equal footing. The frame theory has a parameter λ = c ⁻² that can be either zero or positive, leading to Galilean or Lorentzian geometry. Furthermore, it has a second parameter, Newton’s constant G, that can also be zero or positive, leading to four different theories depending on whether λ and G are zero or positive. Setting G = 0 means that one considers test particles in a fixed background that obeys some curvature constraints. The motivation behind frame theory is to put the 1/c expansion on a firmer mathematical foundation, improving on the work of Dautcourt. For a review of the frame theory, we refer to [29].

Ehlers reviewed Trautman’s work and noticed that the condition in the first equation of (2.2) could be written in other forms. In particular, it can be shown that the following equations are all equivalent formulations of this condition: 0 = R μ ν σ ρ R α β ρ σ h ν β , (2.6) 0 = R μ ν [ σ ρ τ γ ] , (2.7) 0 = R μ ν σ ρ h γ σ . (2.8)

These three conditions are now often referred to as the Ehlers conditions. The main point of these conditions is to restrict the geometry such that the NC gravity in Eq. 2.3 only contains the Newtonian potential (in appropriate coordinates). However, as we know from Dautcourt’s work (and as was also known to Ehlers), these conditions are unnecessary, as this restriction can also be achieved by imposing asymptotic flatness. Ehlers [30] gave a short review of frame theory and examples of Newtonian limits of the well-known GR solutions.

Part of the motivation behind frame theory is the question of whether solutions to Newtonian gravity are extendable (e.g., in the sense of post-Newtonian corrections) to full relativistic solutions of GR. Rendall [31] showed that post-Newtonian corrections are incompatible with asymptotic flatness. The physical reason behind this is that these corrections do not correctly describe the far zone of some non-relativistic matter distribution where gravitational waves dominate. In other words, the post-Newtonian regime corresponding to, for example, a perfect fluid matter source has a finite radius of validity. This effect is noticeable when one goes beyond the first post-Newtonian correction. Lottermoser [32, 33] showed that the constraint equations of GR (in harmonic gauge) admitted a well-defined 1/c expansion (in the sense of a convergent series).

In the current literature on post-Newtonian corrections, which will not be reviewed here (e.g., [34, 35] and references therein), the dominant approach does not consist of expanding the Einstein equations in powers of 1/c and then solving them order by order, which is sometimes called the classic approach. Instead, one formally solves the Einstein equations in harmonic gauge (using what are known as the relaxed Einstein equations ⁶ ) and imposes a boundary condition that leads to an integral equation using a retarded Green’s function. In the Blanchet–Damour approach, ⁷ this integral equation is solved outside the source (i.e., in vacuum) as an expansion in G. In a region containing the non-relativistic source, the integral equation is solved as an expansion in 1/c. The G and 1/c expansions are then matched multipole by multipole in their overlap region using matched asymptotic expansion.

The recent work on the covariant 1/c expansion of GR can be viewed as an attempt to revive the classic approach used in the early days of work on post-Newtonian expansions. It also goes beyond that because it can cover regimes of strong gravity ⁸ and is more flexible regarding the gauge choice one uses. In general, one will have to match the 1/c expansion onto a G expansion and thus find some hybrid of the classical and more modern Blanchet–Damour or Will–Wiseman approaches, which is the aim of the upcoming work [36] (see also [37] for a covariant approach to the post-Newtonian expansion up to 1PN order).

Finally, even though it is not the focus of this review article, we would be remiss not to mention Duval et al.’s work [38]. So far, we have discussed NC geometry either intrinsically or as originating from the 1/c expansion of GR. Duval et al. offered a third perspective by showing that NC geometry can also be obtained from the null reduction of a Lorentzian geometry with a null isometry. They [38] (see [39] for further developments and [40] for a review) considered null reduction of Lorentzian geometry and null uplifts of NC geometry to Lorentzian geometry with a null Killing vector. This is related to the Eisenhart lift [41] of Hamiltonian dynamics as shown in [42]. ⁹ It is also related to the Bargmann algebra [43] because this algebra is the centralizer of the null isometry in the higher-dimensional Poincaré algebra [44], as shown in Eq. 5.4.

Duval et al. [38] discussed uplifts of NC geometries for which dτ = 0 to pp-waves in one dimension higher. This can be generalized to cases where τ ∧ dτ = 0 [45] and even to cases where τ ∧ dτ ≠ 0 [46].

However, as reviewed in Section 5.5, Eq. 2.3 for NC gravity coupled to matter cannot be obtained from null reduction. Despite this shortcoming, it remains useful to adapt this higher-dimensional perspective for many aspects, such as particle motion in a fixed background and more geometrical questions.

The main objects of interest are the nowhere-vanishing “clock” one-form τ _μ and the nowhere-vanishing “spatial” co-metric h ^μν that has a signature (0, 1, …, 1). These objects obey the condition that τ _μ h ^μν = 0. We take the dimension of the underlying spacetime manifold to be (d + 1), so that the spatial co-metric has rank d. We can also define the inverse objects v ^μ and h _μν by demanding that the following relations hold: h μ ρ h ρ ν − τ μ v ν = δ μ ν , v μ h μ ν = 0 , τ μ v μ = − 1 , τ μ h μ ν = 0 . (3.1)

Another way of phrasing these relations among the various objects is by saying that τ _μ τ _ν + h _μν is invertible with inverse v ^ν v ^ρ + h ^νρ . The positive determinant of τ _μ τ _ν + h _μν is denoted by e ², and we can use e as an integration measure. The objects v ^μ and h _μν are defined up to a local Galilean boost, which acts as δ v μ = h μ ν λ ν , δ h μ ν = τ μ λ ν + τ ν λ μ . (3.2)

Here, we require v ^μ λ _μ = 0, and λ _μ transforms under a second Galilean boost transformation as δλ _μ = τ _μ h ^νρ λ _ν λ _ρ so that v ^μ λ _μ = 0 and v ^μ h _μν = 0 are boost invariant under a second-order boost transformation. The exponentiation of the infinitesimal Galilean boost transformation terminates after the second order in λ _μ . The reason why we call these transformations Galilean boosts ¹⁰ will become clear in Section 5. We note that the integration measure e is Galilean boost invariant. Finally, we can introduce frame fields e μ a and e μ a for the degenerate spatial metric and co-metric, which satisfy h μ ν = δ a b e μ a e ν b , h μ ν = δ a b e μ a e ν b , (3.3) where a = 1, …, d are flat frame indices. The integration measure e = det ( τ μ , e μ a ) is the determinant of this spatial vielbein together with the clock one-form.

Consider the case of a symmetric connection that is metric-compatible in the sense that ∇ μ τ ν = 0 , ∇ μ h ν ρ = 0 , (3.4) such that Γ [ μ ν ] ρ = 0 . We can split such a connection as Γ μ ν ρ = Γ ̌ μ ν ρ + C ρ μ ν , (3.5) where C ρ μ ν is symmetric in μ and ν and transforms as a tensor under coordinate transformations, and furthermore, we define Γ ̌ μ ν ρ = − v ρ ∂ μ τ ν + 1 2 h ρ σ ∂ μ h ν σ + ∂ ν h μ σ − ∂ σ h μ ν . (3.6)

As we have assumed Γ [ μ ν ] ρ = 0 in Trautman’s postulate 3 mentioned previously, it follows that we have dτ = 0. Demanding metric compatibility leads to C ρ μ ν τ ρ = 0 , C ν μ λ h λ ρ + C ρ μ λ h ν λ = 0 . (3.7) We can solve the first of these conditions by writing C ρ μ ν = h ρ σ Y σ μ ν , where Y _σμν = Y _σνμ . The second condition tells us that h ν σ h λ ρ + h ρ σ h ν λ Y σ μ λ = 0 . (3.8)

Using completeness on the μ index (i.e., invoking δ μ κ = − τ μ v κ + h μ ρ h ρ κ following Eq. 3.1), we can write Y σ μ ν = − 1 2 τ μ F ν σ − 1 2 τ ν F μ σ + L σ μ ν , (3.9) for some F _μν and some L _σμν = L _σνμ , which is purely spatial (meaning that all contractions with v ^ρ are zero). The factor of −1/2 is there for later convenience. From Eq. 3.8, we then get two equations by contracting with v ^μ and h ^μκ : h ν ρ h λ σ + h ρ σ h ν λ F ν σ = 0 , L σ μ ν = − L ν μ σ . (3.10)

L _σμν must be antisymmetric in its first two indices because L σ μ ν = L σ ν μ = − L μ ν σ = − L μ σ ν . (3.11)

Therefore, we have a tensor L _σμν that is antisymmetric in slots 1 and 2, as well as in slots 1 and 3, and it is symmetric in slots 2 and 3. Such a tensor is zero, as follows from L σ μ ν = − L μ σ ν = L ν σ μ = L ν μ σ , (3.12) which shows that L is symmetric in slots 1 and 3, but that means it must be zero because it is also antisymmetric in slots 1 and 3. Hence, we have C ρ μ ν = − 1 2 h ρ σ τ μ F ν σ + τ ν F μ σ , (3.13) where h ν ρ h λ σ + h ν λ h ρ σ F ν σ = 0 so that F _μν is of the form F _μν = τ _μ X _ν + τ _ν Y _μ + X _μν , where X _μν is purely spatial and antisymmetric. As only h ^ρσ F _νσ appears in the expression for C ρ μ ν , we can, without loss of generality, set Y _μ = −X _μ so that F _μν is antisymmetric. Therefore, we see that Γ μ ν ρ = Γ ̌ μ ν ρ − 1 2 h ρ σ τ μ F ν σ + τ ν F μ σ , (3.14) where Γ ̌ μ ν ρ is given in (3.6). Equation 3.14 is the most general torsion-free NC metric-compatible connection. Unlike in Lorentzian geometry, we see here that this connection is not unique, and its freedom is parametrized by the antisymmetric tensor F _μν [21, 22].

We stress that Γ ̌ μ ν ρ is not invariant under Galilean boosts, and therefore, the covariant derivative whose connection coefficients are given by Γ ̌ μ ν ρ does not form a proper affine connection (although the covariant derivative transforms correctly under general coordinate transformations). For Γ μ ν ρ , as defined in (3.14), to be boost-invariant, F _μν must transform appropriately under Galilean boost transformations. This is possible for a symmetric connection, and we will reach this point in the following paragraphs. With this caveat in mind, we will still use a covariant derivative ∇ ̌ μ with connection coefficients Γ ̌ μ ν ρ and define an associated Riemann curvature R ̌ μ ν σ ρ in the usual way. We refer to Supplementary Appendix SA for our curvature tensor conventions.

For completeness, we briefly comment on the first Trautman condition, corresponding to the first equation of (2.2), even though we will not use it in the following. First, we show that it is equivalent to the Ehlers conditions: 1 R μ ν σ ρ R λ κ ρ σ h ν κ = 0 , (3.15) 2 R μ ν [ σ ρ τ γ ] = 0 , (3.16) 3 R μ ν σ [ ρ h γ ] σ = 0 . (3.17)

We will show that 1) is equivalent to 2) and that 2) is equivalent to 3).

First, we show that 1) implies 2). We project 1) with v ^μ v ^λ . In terms of tangent space indices, this means that the condition becomes R _0abc R _0abc = 0, where R 0 a b c = − v μ e ν a e σ b e ρ c R μ ν σ ρ and where we used R μ ν σ ρ τ ρ = 0 , which follows from the covariant constancy of τ _μ . This is a sum of squares and hence R _0abc . Contracting condition 1) with h ^μλ , we find R _abcd R _abcd = 0 so that R _abcd = 0. From R _0bcd = 0 and R _abcd = 0, it follows that R μ ν σ ρ h σ γ = 0 . The latter equation is equivalent to condition 2). This follows from 0 = R μ ν σ ρ h σ γ h γ κ = R μ ν κ ρ + R μ ν ρ σ v ρ τ κ , where we used completeness.

The other direction, that 2) implies 1), is more straightforward as 2) is equivalent to R μ ν σ ρ h σ γ = 0 , which implies 1). Here, we used again R μ ν σ ρ τ ρ = 0 . To show that 2) is equivalent to 3), all we need is R μ ν σ ρ h γ σ = 0 , which follows from [∇ _μ , ∇ _ν ]h ^ργ = 0. This result implies that R μ ν σ ρ h σ γ = R μ ν σ ρ h γ σ = 0 from which the result follows.

Therefore, what is the purpose of the first Trautman condition? To answer this, we look closer at R μ ν σ ρ h σ γ = 0 . Expressing this in terms of R ̌ μ ν σ ρ and projecting by h ^μα h ^νβ and v ^ν h ^μκ , we find 0 = h μ α h ν β R ̌ μ ν σ ρ h σ γ , (3.18) 0 = h ρ α h γ λ h β κ ∇ ̌ α K λ β − ∇ ̌ λ K α β + 1 2 ∇ ̌ β F α λ , (3.19) where K α β = − 1 2 L v h α β with L v , the Lie derivative along v ^μ . The first equation states that we must always have a flat geometry on constant time slices (these are the surfaces to which τ is the normal one-form). The second equation is a condition on the magnetic part of the field strength F _μν . However, these conditions are too strong. It should only be the equivalent of the Einstein equation that decides what the allowed spaces are. In addition, the first Trautman condition does not follow from the 1/c expansion of Lorentzian geometry.

The relation between the Riemann tensors associated with two connections related via (3.5) is given by R μ ν σ ρ = R ̌ μ ν σ ρ − ∇ ̌ μ C ρ ν σ + ∇ ̌ ν C ρ μ σ − C ρ μ λ C λ ν σ + C ρ ν λ C λ μ σ , (3.20) where on the left-hand side, we have the Riemann tensor associated with ∇ _μ , and on the right-hand side, we have the Riemann tensor associated with ∇ ̌ μ . This gives, for the Trautman condition (2.4), that 0 = h μ γ R μ ν σ ρ = h μ γ R ̌ μ ν σ ρ + 3 4 h μ γ h ρ κ τ ν ∇ ̌ μ F σ κ ] + τ σ ∇ ̌ μ F ν κ ] . (3.21) It can be shown that h μ [ γ R ̌ μ ( ν σ ) ρ ] = 0 . The details can be found in Supplementary Appendix SB. For now, let us see what its consequences are 0 = h μ γ h ρ κ τ ν ∇ ̌ [ μ F σ κ ] + τ σ ∇ ̌ μ F ν κ ] . (3.22) Contracting this equation with v ^ν v ^σ and v ^ν h ^σλ , we see that this is true if and only if we have 0 = ∇ ̌ [ μ F σ κ ] = ∂ [ μ F σ κ ] , (3.23) where the torsion is zero. In other words, the Trautman condition (2.4) leads to a Bianchi identity for F _μν .

We point out that as an alternative to the Trautman condition (2.4), one could also impose h ρ [ κ R μ ν ] σ ρ v σ = 0 , (3.24) whenever Γ μ ν ρ is symmetric because of the following identity: h ρ [ κ R μ ν ] σ ρ v σ = − ∇ [ μ F ν κ ] = − ∂ [ μ F ν κ ] , (3.25) where we used (3.14).

From Eq. 3.23, we see that the second Trautman condition (2.4) is obeyed if we take F _μν = ∂ _μ m _ν − ∂ _ν m _μ with m _μ defined up to a gauge transformation of the form δm _μ = ∂ _μ σ. For the connection (3.14) to be invariant under Galilean boosts, we need m _μ to transform as δm _μ = λ _μ , where we remind the reader that v ^μ λ _μ = 0. Furthermore, in contrast to the first Trautman condition, the second Trautman condition does follow from the 1/c expansion of the following identity when working with a Lorentzian metric and the Levi–Civita connection: g μ [ γ R μ ν ρ σ ] = 0 , (3.26) where in this expression R μ ν ρ σ is the curvature of the Levi–Civita connection. The 1/c expansion of (3.26), assuming that dτ = 0, leads to (2.4), which is equally identically satisfied in NC geometry because the 1/c expansion also tells us that F _μν = ∂ _μ m _ν − ∂ _ν m _μ (see further in Section 3.8).

From now on, we will always take F _μν to be given by ∂ _μ m _ν − ∂ _ν m _μ . For this choice of F _μν , we will denote the connection coefficients by Γ ̄ μ ν ρ , that is, Γ ̄ μ ν ρ = Γ ̌ μ ν ρ − 1 2 h ρ σ τ μ F ν σ + τ ν F μ σ , (3.27) where F _μν = ∂ _μ m _ν − ∂ _ν m _μ . The associated covariant derivatives and curvatures will be denoted by barred quantities.

The total field content of NC geometry is given by the gauge potential m _μ , as well as τ _μ and h _μν . These fields transform under the gauge transformations as δ τ μ = 0 , δ h μ ν = τ μ λ ν + τ ν λ μ , δ m μ = ∂ μ σ + λ μ , (3.28) where v ^μ λ _μ = 0 and where we left out the transformation under diffeomorphisms. The connection (3.14) is inert under these transformations. The symmetry of the connection implies that dτ = 0, and thus, time is absolute.

So far, we have made no statements about dynamics. We now review how standard Newtonian gravity can be formulated in the framework of torsionless NC geometry. Test particles will follow the geodesics of the connection (3.14). In other words, they are described by x ̈ μ + Γ ̄ ν ρ μ x ̇ ν x ̇ ρ = x ̈ μ + Γ ̌ ν ρ μ x ̇ ν x ̇ ρ − h μ σ x ̇ ρ F ρ σ = 0 , (3.29) where the dots denote derivatives with respect to λ, the affine parameter along the geodesic. We set τ μ x ̇ μ = 1 , which implies that the τ _μ contraction of this equation is trivially satisfied. This geodesic equation follows from the action S = m ∫ d λ h μ ν x ̇ μ x ̇ ν 2 τ ρ x ̇ ρ − m μ x ̇ μ . (3.30)

This action has worldline reparametrization invariance δλ = ξ(λ) and δ x μ = ξ ( λ ) x ̇ μ , which can be used to fix τ μ x ̇ μ = 1 . The fact that m appears as an overall parameter in this particle action, so that inertial and gravitational masses are equal, manifests the equivalence principle. The time component of m _μ is the Newtonian potential.

Following Trautman’s postulate 5, the equations corresponding to Newtonian gravity should take the following covariant form in terms of the NC geometry: R ̄ μ ν = 8 π G d − 2 d − 1 ρ τ μ τ ν , , d τ = 0 , (3.31) where R ̄ μ ν = R ̄ μ σ ν σ is the Ricci curvature associated with (3.27), where ρ is the mass density of a non-relativistic matter source and (d + 1) is the total spacetime dimension. This corresponds to the Poisson equation of Newtonian gravity written in an arbitrary frame. Eq. 3.31 follows from an action principle obtained from the 1/c ² expansion of the Einstein–Hilbert (EH) action coupled to the action of a massive point particle [10], as shown in Section 5.5.

Rewriting the equations for NC gravity (3.31), which are expressed in terms of the Γ ̄ μ ν ρ connection to a set of equations that are expressed in terms of the Γ ̌ μ ν ρ connection, the equations for NC gravity can be written as (where we have contracted (3.31) with v ^μ and h ^μν ) h μ ρ h ν σ R ̌ ρ σ = 0 , (3.32) h μ ρ v σ R ̌ ρ σ = − 1 2 e − 1 ∂ λ e h μ ρ h λ σ F ρ σ , (3.33) v ρ v σ R ̌ ρ σ = − e − 1 ∂ ρ e v ν h ρ σ F ν σ − 1 4 h μ ν h ρ σ F μ ρ F ν σ + 8 π G d − 2 d − 1 ρ . (3.34)

In NC gravity, this should be supplemented with the condition dτ = 0. The left-hand side is pure geometric data, and the right-hand side depends entirely on the “electric” and “magnetic” field strength components of F _μν . We will see in the following sections that the divergence of the electric field strength in the third equation, that is, e − 1 ∂ ρ e v ν h ρ σ F ν σ , is what gives rise to Newton’s law of gravity. To arrive at Newtonian gravity, we need to somehow get rid of the magnetic field strength term, h ^μν h ^ρσ F _μρ F _νσ , in Eq. 3.34. This is the rationale behind the first Trautman condition. However, it was later realized that this condition is not necessary as one can argue that the magnetic field strength has to be zero as a result of a boundary condition that states that h ^μν h ^ρσ F _μρ has to vanish at infinity. This latter condition follows from the 1/c expansion of an asymptotically flat metric. Therefore, this is a more minimal approach to recovering Newtonian gravity.

In order to recover Newtonian gravity in its usual form, it is unavoidable to talk about gauge fixing the NC gauge symmetries (3.28) and diffeomorphisms. A covariant definition of a locally flat NC geometry could be to simply require R ̄ μ ν σ ρ = 0 . This condition is invariant under the Galilean boost and σ-gauge transformations of Eq. 3.28. However, in practice, only τ _μ and h _μν are often treated as geometric fields, whereas m _μ is interpreted as a force field. This is not a covariant distinction and largely results from historical bias. For example, one can always use (3.28) to gauge away m _μ entirely, as shown in [47]; see [48] for some examples.

A common gauge fixing that can always be made when dτ = 0 is as follows: first, we partially fix diffeomorphisms so that τ = dt, where t is our time coordinate. In this class of coordinate systems, the boosts act as δh _ti = λ _i , where the latter is an arbitrary one-form in d spatial dimensions. We can thus completely fix the Galilean boost symmetry by demanding that h _ti = 0 in these coordinates. We then find from v ^μ h _μν = 0 and v ^μ τ _μ = −1 that v ^t = −1 and h _tt = 0 and that v ⁱ = 0, where h _ij is invertible with inverse h ^ij . Using τ _μ h ^μν = 0, we find h ^tt = 0 = h ^ti . To summarize, we can always go to a gauge in which τ = d t , h μ ν d x μ d x ν = h i j d x i d x j , v = − ∂ t , h μ ν ∂ μ ∂ ν = h i j ∂ i ∂ j , (3.35) and where m _μ is completely arbitrary. This gauge choice still has residual gauge transformations acting on it. Demanding that the geometry on constant time slices is flat leads to the condition that h α μ h β ν h γ σ R ̌ μ ν σ ρ = 0 (which is invariant under Galilean boosts). In our partially fixed gauge (3.35), this amounts to demanding that the Riemann tensor of the Riemannian metric h _ij is zero. We can fix diffeomorphisms further so that locally h _ij = a ²(t)δ _ij for some function a(t). This function plays an important role in Newtonian cosmology. We can perform a coordinate transformation of the form x′ ⁱ = a(t)x ⁱ and t′ = t followed by a finite Galilean boost so that in the primed coordinate system h μ ν ′ d x ′ μ d x ′ ν = δ i j d x ′ i d x ′ j and τ = dt′. This transformation will affect the m _μ connection, but the point here is to fix τ _μ and h _μν as much as possible. Thus, without loss of generality, we can set a(t) = 1. Hence, we will continue by working with τ = d t , h μ ν d x μ d x ν = δ i j d x i d x j , v = − ∂ t , h μ ν ∂ μ ∂ ν = δ i j ∂ i ∂ j . (3.36)

In this case, all the non-trivial information about the NC geometry is in m _μ , which remains fully arbitrary at this point. In this gauge and with these choices, Eqs (3.32)–(3.34) reduce to 0 = ∂ i F i j , (3.37) 0 = ∂ i F t i − 1 4 F i j F i j + 8 π G d − 2 d − 1 ρ . (3.38) The first of these two equations is solved by F _ij = ɛ _ijk ∂ _k F. The Bianchi identity ∂ i F j k = 0 (which follows from the Trautman condition) then tells us that F is harmonic on flat space (without any sources). By using Liouville’s theorem, this function must be constant to be regular everywhere (including infinity); hence, we conclude that F _ij = 0. We can gauge fix m _i = 0, using the freedom to transform m _μ as δm _μ = ∂ _μ σ. Now, the NC gravity equations simplify to the well-known Poisson equation: ∇ 2 Φ N = 8 π G d − 2 d − 1 ρ , (3.39) where we defined m _t = Φ _N , the Newtonian potential. Therefore, we see that the covariant Eq. 3.31 reproduces the usual form of Newtonian gravity in an appropriate coordinate system.

We very briefly review how Trautman’s definition of NC gravity follows from the 1/c expansion of GR. We will have much more to say about the 1/c expansion in Section 6, so we will keep it brief. The following is essentially Dautcourt’s work [23, 24]. We will only consider even powers of 1/c ² and assume analyticity in 1/c ². The metric expands as g μ ν = − c 2 τ μ τ ν + h μ ν − τ μ m ν − τ ν m μ + O c − 2 , (3.40) where h _μν has signature (0, 1, …, 1) and τ _μ τ _ν + h _μν is invertible. The Christoffel connection expands as Γ μ ν ρ = Γ ̄ μ ν ρ + O c − 2 , (3.41) where Γ ̄ μ ν ρ is given in Eq. 3.27, but only provided we set dτ = 0 by hand. If we also expand the diffeomorphisms parameter Ξ μ = ξ μ + c − 2 ζ μ + O ( c − 4 ) , then m _μ transforms as δm _μ = ∂ _μ σ with σ = τ _μ ζ ^μ under the subleading diffeomorphisms with parameter ζ ^μ , again only provided that dτ = 0. Finally, the combination h _μν − τ _μ m _ν − τ _ν m _μ is invariant under the Galilean boost transformation with parameter λ _μ discussed previously. We recover the NC fields and their gauge properties as discussed previously but only under the assumption that dτ = 0. The 1/c expansion of matter will be discussed in later sections, but an important observation is that one can only obtain the NC geodesic Equation 3.29 from a 1/c expansion if dτ = 0. Then, the 1/c expansion of the Einstein equations coupled to the energy-momentum tensor of a massive point particle leads to (3.31). If g _μν is asymptotically flat, it follows that m _μ (in the limit r → ∞) is at most pure gauge at spatial infinity. This is why, in (3.38), F _ij is non-singular at infinity.

As we will review momentarily, one can obtain Lorentzian geometry by applying a gauging procedure to the Poincaré algebra. Similarly, type I TNC geometry can be obtained from a gauging of the Bargmann algebra, which encodes its local symmetries. In this section, we first show how the Bargmann algebra can be obtained from a contraction and a null reduction of the Poincaré algebra. These two derivations will be mirrored at the level of the point particle and gravity actions later.

First, we consider an İnönü–Wigner contraction of the Poincaré algebra trivially extended with a U(1) generator that commutes with the entire Poincaré algebra, which we denote by Q. As shown in the following, this generator is associated with electromagnetic coupling. We work in (d + 1) spacetime dimensions and use the following conventions for the Poincaré algebra: M A B , M C D = η A C M B D − η B C M A D + η B D M A C − η A D M B C , (5.1a) M A B , P C = η A C P B − η B C P A . (5.1b)

Here, P _A are translation generators and M _AB = −M _AB are the Lorentz and rotation generators. Now, we consider a spacetime split of the (d + 1)-dimensional algebra indices A = (0, a), where a = 1, …, d is a spatial index, and we define the generators H = c P 0 + Q , N = 1 c P 0 , G a = 1 c M 0 a , J a b = M a b . (5.2)

So far, this is just a change of basis. However, if we now take the limit c → ∞, we end up with an inequivalent algebra whose non-zero commutation relations are J a b , J c d = δ a c J b d − δ b c J a d + δ b d J a c − δ a d J b c , (5.3a) J a b , P c = δ a c P b − δ b c P a , J a b , G c = δ a c G b − δ b c G a , (5.3b) G a , H = − P a , G a , P b = − δ a b N . (5.3c)

This is the Bargmann algebra. For N = 0, we recover the Galilei algebra, which contains time translations H, spatial translations P _a , and rotations J _ab , as well as the Galilean boosts G _a . For N ≠ 0, we obtain the centrally extended Galilei algebra known as the Bargmann algebra. The central extension N, as shown later, corresponds to the gauge potential m _μ that was introduced previously.

Another way to obtain the algebra (5.3) is via a null reduction of the Poincaré algebra in (d + 2) dimensions [43]. If we consider all generators of this algebra that commute with the null translation generator N = P + = ( P 0 + P d + 1 ) / 2 , we get P a , H = P − = P 0 − P d + 1 / 2 , J a b = M a b , G a = M + a = M 0 a + M d + 1 a / 2 , (5.4) which forms a subalgebra corresponding to the Bargmann algebra (5.3a). Note that the higher-dimensional Lorentz boosts M _−a and M ₊₋ do not commute with H and therefore do not enter this subalgebra.

As was first realized in [3], type I TNC geometry can be obtained by gauging the Bargmann algebra (5.3b). This section reviews this gauging construction, including its generalization to non-zero torsion [49]. ¹¹ Before that, we briefly review the well-known procedure for obtaining Lorentzian geometry from a gauging of the Poincaré algebra. Finally, we also show how type I TNC geometry can be obtained from a null reduction of higher-dimensional Lorentzian geometry.